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For any type of formation F , we denote by
probability of F being achieved by river by your own hand and by
probability that at least one opponent will achieve something higher
than F, if you will achieve F.
We will abbreviate the types of Texas Hold'em formations as follows: 1p –
one pair; 2p – two pairs, 3k – three of a kind; st
– flush; fh – full house; and 4k – four of a kind.
hand we mean the card configuration of the entire board
(community board and your own hole cards), along with the number of
your opponents in play at the moment of analysis. This holds for any
For any hand and type of formation F, we
call the pair
strength vector of that hand with respect to F.
Community cards: (8♦ 5♠ 2♣) (flop stage)
- Your own hole cards:
For full house (fh), we
must consider all possible reconstitutions of the community board in
the river stage, in
the assumption you achieve a full house by river, then calculate the
q probability for each case, and then using some probability
formulas for obtaining the overall probability. This process of
calculation is described in detail in the chapter Evaluating the
strength of a hand of the book
Texas Hold’em Poker Odds for Your Strategy, with Probability-Based
Hand Analyses). The
strength vector of this hand with respect to full house is (2%, 41%)
and reads as follows:
have about 2% chance for achieving a full house by river; in case
you do achieve it, your opponents have about 41% chance of beating
is a weak hand with respect to full house: only 2% chance of
achieving it and then 41% chance of loosing (59% of winning) with
it. Thus, you might consider other inferior formations to analyze.
the strength vector of a hand with respect to F, we call the
strength indicator of that hand with respect to F.
In our previous example, the strength indicator of the hand with respect
to full house is .
also a probability, namely the probability that you will achieve a
formation of type F by river and none of your opponents will
achieve something higher, if you achieve it.
If we were to consider all types
of formations (from one pair to four of a kind) and do the same
calculations for each of them, we would end up with seven strength
vectors, one for each type of formation. These seven vectors would give
us the whole image of the strength of that hand. In fact, they form a 2
7 matrix of probabilities.
each hand, we call the matrix
strength matrix of that hand.
On the first row of the strength matrix are the probabilities of the own
hand achieving the various types of formations by river. On the second
row are the corresponding probabilities that your opponents (at least
one) will achieve something higher than you by river, if you will
achieve that expected type of formation. Each column corresponds to a
type of formation (1p, 2p, 3k, s, fl, fh, 4k).
Each hand has
an unique associated strength matrix, whose elements are calculable
manually or by software program.
Conventions. The following conventions were established in order to
compute a strength matrix more easily:
flush has not been included on the last column as the highest type of
formation. In fact, it is included in the flush type. When you hit a
straight flush, there is nothing to analyze - you should put all in.
We replace by 0 the
elements of any column of a type of formation that is included at the
moment of analysis or will be included – should it occur – by river in a
superior type of formation. By this convention, the ignored type of
formation F gets a null probability of being achieved
consequently a null probability for the opponents to beat it if it
Thus, it practically comes out of the hand analysis. For
instance, the strength matrix
associated to a hand in turn stage where you have a full house, has the
In this case,
one pair, two pairs, and three of a kind cannot be achieved as the full
house includes and cancels them, and straight is impossible since you
have at maximum two cards of it. Thus full house and four of a kind
remain to be dealt with.
Interpretation. As the strength of a poker hand can only be expressed
through mathematical probabilities of final events, the strength matrix
is the most adequate object to picture such strength. When we evaluate
the strength of a hand by interpreting its strength matrix, we actually
assume a scale on which to place that strength, and implicitly a
relation order over all possible hands. This is what “how strong is it”
means in mathematical terms. Assuming we have the strength matrix of a
hand we want to analyze, how will we actually interpret it? The rough
rule is: The higher the p-probabilities and the lower the q-probabilities,
the stronger the hand. However, if we consider the p row, it is
better for the p-probabilities to be higher in the second part
than in the first, as the second part corresponds to the most valuable
achievements. In fact, a high value of a p-probability for only
one type of formation of the second part (s, fl, fh, or 4k) may be
sufficient for considering the hand strong enough for aggressive
raising, as example. Having high values of the p-probabilities in
the first part (for 1p, 2p, or 3k) is not a positive factor in the
hand’s strength, since consequently we will have lower values in the
second part, which means that the most valuable formations are unlikely
to be achieved. This happens because the sum of the p-probabilities
has an upper bound. The strength matrix cannot be interpreted only by
the p-row. The q-probabilities are also important, as they
can raise or temper the trust one may have in the corresponding p-probabilities
with respect to the outcome of the decision made basing on them.
- For example, if
a strength vector for a type of formation shows (0.55, 0.73), one
may not rely on that good p-probability of over 50%, as long
as the opponents may beat him/her with a q-probability of
73%, which is relatively high. Conversely, if a strength vector
shows (0.17, 0.08), although 17% is not that much for achieving that
type of formation, one may consider it worth that risk, as the
opponents have only an 8% chance of beating him/her.
- Of course, for a
complete analysis, the entire matrix (all strength vectors) should
be evaluated and interpreted. That is because when a strength vector
shows non-favorable probabilities, one may look for alternatives
among the other types of formations and these other strengths have a
cumulative effect toward that hand’s strength evaluation.
- In the river stage, the board will no
longer be subject to reconstitution and the probabilities of the
types of formations to be achieved by the own hand (which are sure
in this case) are 1 or 0 (1 for those achieved and 0 for the rest).
Thus, the p-row of a strength matrix of a river hand will
contain only 1’s and 0’s, which of course will render the
calculations for completing it much easier. Here is an example:
Community cards: (2♦ 2♣ 4♥ 5♥ 6♦)
- Your own
hole cards: (2♠ 3♥)
You achieved three
of a kind and straight.
The strength matrix of this hand is
Looking only in the straight column, for a straight achieved and only
11% chances of your opponents having something higher, one may consider
this hand strong enough to put all in.
The strength indicator.
exists a way of aggregating the data of a strength matrix in order for
the strength to be interpreted through a single value as indicator and
not through 14 values. Since every
type of formation counts differently in what we call the strength of the
hand expressed through probability (the unit strength of two pairs must
weight less than the unit strength of a straight, for instance; that is,
roughly speaking, that 1% chances of achieving two pairs weights less
than 1% chances of achieving a straight), this indicator must be a
weighted mean of the strength indicators of the hand with respect to
each type of formation.
For any hand, we call the number
where the weights
predefined in the next table, the strength
indicator of that hand.
Type of formation
three of a kind
four of a kind
The weights that
show in the weight column are the normalized values of the inverses of
the probabilities of the types of formations occurring in a random
previous example, the strength indicator is
Other may refer to the strength of a poker hand in various ways, from which
the mostly used are statistical. In their view, a hand is strong
depending upon how often it has won in the past, when and where it
occurred. In statistical terms, they assign the quality of being weak or
strong in various degrees to a hand referring to relative frequency
instead of probability, while this latter is a limit of relative
frequencies and stands as the most objective measure of possibility and
degree of belief in the occurrence of an event. All kinds of software
called “poker odds calculator” help them in making this assignment. Read
Odds: Partial simulations vs. compact formulas
how the results from such software differ from the mathematical
probabilities and implicitly from the strength indicator defined above.
hand analyses. For a given concrete hand, completing its strength
matrix by manual calculations is a laborious job.
With regard to the practical
analysis of the hands, many times we should focus only on a portion of
the types of formations expected to be achieved, sometimes only on one
type of formation. This reduction may due to either the particular
configuration of the board, which makes impossible the achievement of
other types of formations, or the relevance of the obtained information
with respect to the threshold of afforded risk.
If you run
such a reduced analysis, in fact you partially complete the strength
matrix of that hand.
There are reductions and
approximations for the probability calculations applied in Hold’em and
these can be also applied to the evaluations of the strength of a hand.
As the strength matrix of a hand gives an image more relevant to what
that hand can offer to its owner than the strength indicator, a reduced
hand analysis would mean the interpretation of the partially completed
strength matrix. Of course, the parts of that matrix to be computed and
interpreted should be chosen by combined criteria of relevance and ease
of calculations. Analysis of particular hands may not even require
reference to their strength matrices, but can be run directly on those
particular card configurations, by using all the reduction methods
here for some probability-based
of concrete Hold'em hands.
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All Hold'em odds, for all gaming situations, and all the math behind the game, are covered in the book
Your Strategy, with Probability-Based Hand Analyses. The
book holds all you need to now for using probability
in your strategy, including examples and a relevant collection of probability-based analyses on
concrete Hold'em hands. See the Books
section for details.